A Conjecture on Integer Arithmetic which Implies that there is an Algorithm which to each Diophantine Equation Assigns an Integer which is Greater than the Heights of Integer (Non-negative Integer, Rational) Solutions, if these Solutions Form a Finite Set

Apoloniusz Tyszka , Maciej Sporysz , Agnieszka Peszek

Abstract

We conjecture that if a system S ⊆ {xi = 1, xi + xj = xk , xi · xj = xk : i, j, k ∈ {1, . . . , n}} has only finitely many solutions in integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies |x1|, . . . , |xn| ≤ 2 2 n−1 . The conjecture implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set. We present a MuPAD code whose execution never terminates. If the conjecture is true, then the code sequentially displays all integers n ≥ 2. If the conjecture is false, then the code sequentially displays the integers 2, . . . , n − 1, where n ≥ 4, the value of n is unknown beforehand, and the conjecture is false for all integers m with m ≥ n.
Author Apoloniusz Tyszka (FoPaPE)
Apoloniusz Tyszka,,
- Faculty of Production and Power Engineering
, Maciej Sporysz (FoPaPE)
Maciej Sporysz,,
- Faculty of Production and Power Engineering
, Agnieszka Peszek (FoPaPE)
Agnieszka Peszek,,
- Faculty of Production and Power Engineering
Journal seriesInternational Mathematical Forum , ISSN 1312-7594 , (0 pkt)
Issue year2013
Vol8
No1
Pages39-46
Publication size in sheets0.5
Keywords in Englishcomputable upper bound for the heights of integer (non-negative integer, rational) solutions of a Diophantine equation, Diophantine equation
URL http://arxiv.org/pdf/1109.3826.pdf
Internal identifierWIPiE/2013/89
Languageen angielski
Score (nominal)5
Score sourcejournalList
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